Reducibility of Self-Adjoint Linear Relations and Application to Generalized Nevanlinna Functions

We present necessary and sufficient conditions for the reducibility of a self-adjoint linear relation in Krein space. Then generalized Nevanlinna function Q represented by A Pontryagin space can be decomposed means reducing subspaces A. The sum two functions $$ {Q}_i\in {N}_{\kappa_i}\left(\mathcal{H}\right) , i = 1, 2, minimally triplets \left({\mathcal{K}}_i,{A}_i,{\Gamma}_i\right) is also studied. For this purpose, we create model \left(\overset{\sim }{\mathcal{K}},\overset{\sim }{A},\overset{\sim }{\Gamma}\right) to represent := Q1 + Q2 terms . By using model, κ κ1+ κ2 are proved analytic form. Finally, explain how degenerate Jordan chains representing affect decomposition corresponding Q.